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Reciprocal Specific Relative Entropy between Continuous Martingales

Julio Backhoff, Xin Zhang

TL;DR

The paper introduces the reciprocal specific relative entropy $\mathfrak{h}(\mathbb Q \| \mathbb W)$ as a divergence between continuous martingales and Brownian motion, defined via the instantaneous quadratic variation. It demonstrates that $\mathfrak{h}$ arises as the derivative of the specific $p$-Wasserstein divergence at $p=2$ and as the continuum limit of trinomial-model relative entropies, while linking it to a reciprocal form of the forward entropy through a time-change. Focusing on win-martingale calibration, the authors solve the optimization problem and prove that the unique minimizer is the scaled neutral Wright-Fisher diffusion, with an explicit value function solving a Hamilton-Jacobi-Bellman equation. This result connects Martingale Optimal Transport, stochastic control, and population genetics diffusion, and suggests natural extensions to multidimensional settings and alternative entropy notions. The work provides a principled, entropy-regularized criterion for model selection in prediction markets and martingale calibration.

Abstract

We introduce a novel notion of divergence between continuous martingales; the reciprocal specific relative entropy. First, we motivate this definition from multiple perspectives. Thereafter, we solve the reciprocal specific relative entropy minimization problem over the set of win-martingales (used as models for prediction markets Aldous (2013)). Surprisingly, we show that the optimizer is the renowned neutral Wright-Fisher diffusion. We also justify that this diffusion is in a sense the most salient win-martingale, since it is uniquely selected when we suitably perturb the degenerate martingale optimal transport problem of variance minimization.

Reciprocal Specific Relative Entropy between Continuous Martingales

TL;DR

The paper introduces the reciprocal specific relative entropy as a divergence between continuous martingales and Brownian motion, defined via the instantaneous quadratic variation. It demonstrates that arises as the derivative of the specific -Wasserstein divergence at and as the continuum limit of trinomial-model relative entropies, while linking it to a reciprocal form of the forward entropy through a time-change. Focusing on win-martingale calibration, the authors solve the optimization problem and prove that the unique minimizer is the scaled neutral Wright-Fisher diffusion, with an explicit value function solving a Hamilton-Jacobi-Bellman equation. This result connects Martingale Optimal Transport, stochastic control, and population genetics diffusion, and suggests natural extensions to multidimensional settings and alternative entropy notions. The work provides a principled, entropy-regularized criterion for model selection in prediction markets and martingale calibration.

Abstract

We introduce a novel notion of divergence between continuous martingales; the reciprocal specific relative entropy. First, we motivate this definition from multiple perspectives. Thereafter, we solve the reciprocal specific relative entropy minimization problem over the set of win-martingales (used as models for prediction markets Aldous (2013)). Surprisingly, we show that the optimizer is the renowned neutral Wright-Fisher diffusion. We also justify that this diffusion is in a sense the most salient win-martingale, since it is uniquely selected when we suitably perturb the degenerate martingale optimal transport problem of variance minimization.
Paper Structure (7 sections, 6 theorems, 61 equations)

This paper contains 7 sections, 6 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Reciprocal specific relative entropy
  4. Limit of the specific $p$-Wasserstein divergence
  5. Limit of Trinomial Models
  6. Reversing the roles of $\mathbb Q$ and $\mathbb W$
  7. Win-martingale optimization

Key Result

Lemma 2.1

We always have If an optimizer $\mathbb Q^*$ of eq:MOTre satisfying $\mathbb E_{\mathbb Q^*} \left[ \int_0^1 |\sigma_t|^{2+\epsilon} \, dt\right]$ for some $\epsilon \in (0,1)$ exists, then

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Definition 3.1
  • Lemma 3.1
  • proof
  • ...and 6 more